An Essay towards Solving a Problem in the Doctrine of Chances¶
Bayes, T. (1763). An Essay towards Solving a Problem in the Doctrine of Chances. Philosophical Transactions of the Royal Society of London, 53, 370-418.
Cited by¶
8 citations across 8 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Bayesian Updating
- (1) **Bayesian updating is the systematic process of revising a probability distribution over possibilities — the prior — by combining it with the likelihood of new evidence given each possibility, producing a revised posterior distribution.** (2) Mathematically, posterior ∝ prior × likelihood (Bayes' theorem
This source(Communicated posthumously by Richard Price.) Founding statement of inverse-probability reasoning / Bayes' theorem. SUPPORTS marker 007 (posterior ∝ prior × likelihood).
- (1) **Bayesian updating is the systematic process of revising a probability distribution over possibilities — the prior — by combining it with the likelihood of new evidence given each possibility, producing a revised posterior distribution.** (2) Mathematically, posterior ∝ prior × likelihood (Bayes' theorem
- Conditional Probability
- Two further facts come along for the ride: conditional independence (\(P(A \mid B,C) = P(A \mid C)\)), the structural primitive that makes graphical models and large-scale inference tractable by declaring some conditioning irrelevant given other conditioning; and Bayes' rule (\(P(A \mid B) = P(B \mid A)\,P(A)/P(B)\)), the algebraic relation that inverts the direction of conditioning
This sourceThe original statement of the rule inverting the direction of conditioning, P(A|B) = P(B|A)P(A)/P(B).
- Two further facts come along for the ride: conditional independence (\(P(A \mid B,C) = P(A \mid C)\)), the structural primitive that makes graphical models and large-scale inference tractable by declaring some conditioning irrelevant given other conditioning; and Bayes' rule (\(P(A \mid B) = P(B \mid A)\,P(A)/P(B)\)), the algebraic relation that inverts the direction of conditioning
- Evidence
- The inference rule is Bayes' theorem: the posterior \(P(D\mid +) = \frac{P(+\mid D)\,P(D)}{P(+)}\), which makes the prime's defeasibility/weight-bearing invariant exact — a positive test does not deduce disease, it shifts a prior to a graded posterior.
This sourceOriginal statement of the inverse-probability rule underlying Bayesian updating of a hypothesis on observed evidence.
- The inference rule is Bayes' theorem: the posterior \(P(D\mid +) = \frac{P(+\mid D)\,P(D)}{P(+)}\), which makes the prime's defeasibility/weight-bearing invariant exact — a positive test does not deduce disease, it shifts a prior to a graded posterior.
- False Positive Paradox
- The false positive paradox is the structural fact that when a binary detector is applied to a population in which the target condition is rare, most of the positives it flags will be wrong — even when the detector has high sensitivity and high specificity.
This sourceOriginal statement of Bayes' rule, by which posterior odds equal prior odds times the likelihood ratio — the arithmetic underlying the base-rate dependence of positive predictive value.
- The false positive paradox is the structural fact that when a binary detector is applied to a population in which the target condition is rare, most of the positives it flags will be wrong — even when the detector has high sensitivity and high specificity.
- Inversion
- The Bayesian inversion pattern
This source(Posthumous publication communicated by Richard Price.) Founding text of inverse-probability reasoning that becomes the Bayesian interpretation, mechanizing the update of prior probabilities by conditioning on observed evidence.
- The Bayesian inversion pattern
- Probability
- Computer science and machine learning use probability for randomized algorithms, probabilistic graphical models, Bayesian inference, Monte Carlo methods, information theory (Shannon entropy as the expected log-probability of a code), and reinforcement learning under uncertain dynamics. Decision-making and economics rest on expected utility theory, risk pricing, insurance, portfolio theory, and game theory with mixed strategies — most of the structure traceable to Bayes' (1763) originating insight on inverse probability.
This source(Posthumous publication communicated by Richard Price.) Founding text of inverse-probability reasoning that becomes the Bayesian interpretation, mechanizing the update of prior probabilities by conditioning on observed evidence.
- Computer science and machine learning use probability for randomized algorithms, probabilistic graphical models, Bayesian inference, Monte Carlo methods, information theory (Shannon entropy as the expected log-probability of a code), and reinforcement learning under uncertain dynamics. Decision-making and economics rest on expected utility theory, risk pricing, insurance, portfolio theory, and game theory with mixed strategies — most of the structure traceable to Bayes' (1763) originating insight on inverse probability.
Domain-specific¶
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